Principal $G$-bundles with parabolic structure over a normal crossing divisor are defined along the line of the interpretation of the usual principal $G$-bundles as functors from the category of representations, of the structure group $G$, into the category of vector bundles, satisfying certain axioms. Various results on principal bundles are extended to the more general context of principal bundles with parabolic structures, and also to parabolic $G$-bundles with Higgs structure. A simple construction of the moduli space of parabolic semistable $G$-bundles over a curve is given, where $G$ is a semisimple linear algebraic group over $C$.

We construct local splitting families of hyperelliptic pencils so that the original complicated degenerate fiber decomposes into several simple degenerate fibers. In some sense, our trial is a generalization to hyperelliptic curves of arbitrary genus of Moishezon's construction for families of elliptic curves. Moreover, we study certain invariants of degenerate fiber germs.

Let ${\mathcal O}$ be a nilpotent orbit in $\mathfrak{so}(p,q)$ under the adjoint action of the full orthogonal group ${\rm O}(p,q)$. Then the closure of ${\mathcal O}$ (with respect to the Euclidean topology) is a union of ${\mathcal O}$ and some nilpotent ${\rm O}(p,q)$-orbits of smaller dimensions. In an earlier work, the first author has determined which nilpotent ${\rm O}(p,q)$-orbits belong to this closure. The same problem for the action of the identity component ${\rm SO}(p,q)^0$ of ${\rm O}(p,q)$ on $\mathfrak{so}(p,q)$ is much harder and we propose a conjecture describing the closures of the nilpotent ${\rm SO}(p,q)^0$-orbits. The conjecture is proved when $\min(p,q)\le7$. Our method is indirect because we use the Kostant-Sekiguchi correspondence to translate the problem to that of describing the closures of the unstable orbits for the action of the complex group ${\rm SO}_p({\bf C})\times{\rm SO}_q({\bf C})$ on the space $M_{p,q}$ of complex $p\times q$ matrices with the action given by $(a,b)\cdot x=axb^{-1}$. The fact that the Kostant--Sekiguchi correspondence preserves the closure relation has been proved recently by Barbasch and Sepanski.

Let $(M,g_o)$ be a complete, noncompact Riemannian manifold with a pole, and let $g=fg_o$ be a conformally related metric. We obtain conditions on the curvature of $g_o$ and on $f$ under which the Laplacian on $p$-forms on $(M,g)$ has no eigenvalues.

We analyze the relationship between abelian fourfolds of Weil type and Hodge structures of type K3, and we extend some of these correspondences to the case of arbitrary dimension.

We study the asymptotic behavior of the quadratic means on spheres of increasing radius of the Fourier transforms of characteristic functions of sets with smooth boundary. Beside to give different proofs of known Euclidean results, we also consider the non-Euclidean Fourier transform on the two dimensional sphere and the hyperbolic disk.